The Astrophysical Journal, 702:649–659, 2009 September 1 doi:10.1088/0004-637X/702/1/649C© 2009. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

DYNAMICAL STRUCTURE OF VISCOUS ACCRETION DISKS WITH SHOCKS

Santabrata Das1, Peter A. Becker

2,5, and Truong Le

3,41 Korea Astronomy and Space Science Institute, 61-1, Hwaam Dong, Yuseong-Gu, Daejeon 305 348, South Korea; sbdas@kasi.re.kr

2 Department of Computational and Data Sciences, George Mason University, Fairfax, VA 22030-4444, USA; pbecker@gmu.edu3 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA; tle@stsci.edu

4 Computer Sciences Corporation, 45154 Underwood Lane, Sterling, VA 20166-9514, USAReceived 2008 May 22; accepted 2009 June 29; published 2009 August 14

ABSTRACT

We develop and discuss global accretion solutions for viscous advection-dominated accretion flow (ADAF) diskscontaining centrifugally supported isothermal shock waves. The fact that such shocks can exist at all in ADAFdisks is a new result. Interestingly, we find that isothermal shocks can form even when the level of viscous dis-sipation is relatively high. In order to better understand this phenomenon, we explore all possible combinationsof the fundamental flow parameters, such as specific energy, specific angular momentum, and viscosity, to obtainthe complete family of global solutions. This procedure allows us to identify the region of the parameter spacewhere isothermal shocks can exist in viscous ADAF disks. The allowed region is maximized in the inviscid case,and it shrinks as the level of viscous dissipation increases. Adopting the canonical value γ = 1.5 for the ratio ofspecific heats, we find that the shock region disappears completely when the Shakura–Sunyaev viscosity parameterα exceeds the critical value ∼0.27. This establishes for the first time that steady ADAF disks containing shocks canexist even for relatively high levels of viscous dissipation. If an isothermal shock is present in the disk, it wouldhave important implications for the acceleration of energetic particles that can escape to power the relativistic jetscommonly observed around underfed, radio-loud black holes. In two specific applications, we confirm that thekinetic luminosity lost from the disk at the isothermal shock location is sufficient to power the observed relativisticoutflows in M87 and Sgr A∗.

Key words: accretion, accretion disks – black hole physics – galaxies: jets – hydrodynamics

1. INTRODUCTION

The accretion of matter onto compact objects is the funda-mental mechanism powering a variety of high-energy astrophys-ical sources, such as low-mass X-ray binaries, massive blackholes, and active galactic nuclei. Shakura & Sunyaev (1973)and Novikov & Thorne (1973) laid the foundation for our under-standing of these sources by developing the first physical modelsfor geometrically thin accretion disks. These early models didnot treat the pressure and advection terms in the conservationequations correctly, and furthermore no attempt was made to sat-isfy the inner boundary conditions at the event horizon. Instead,the disk was simply terminated at the marginally stable orbit.The inner regions of these disks were shown to be viscouslyand thermally unstable by Lightman & Eardley (1974), and themodel was subsequently improved by Paczynski & Bisnovatyi-Kogan (1981) and Muchotrzeb & Paczynski (1982) who in-corporated advection and pressure effects into their models.Following a similar approach, the general global solution foraccretion in the hydrodynamic limit, including advection, vis-cosity, and thermal effects, was obtained by Chakrabarti (1990,1996).

1.1. Advection-Dominated Accretion

The advection-dominated accretion flow (ADAF) model re-mains a central paradigm in contemporary studies of blackholeaccretion as a possible explanation for the dynamical structureof sources with significantly sub-Eddington accretion rates (seeYuan 2007 for a review). The gas in the inner regions of suchdisks is quite tenuous, and therefore the plasma radiates very

5 Also at: Department of Physics and Astronomy, George Mason University,Fairfax, VA 22030-4444, USA.

inefficiently, leading to high temperatures. At large radii, thehot ADAF inner region is expected to merge with a cool, geo-metrically thin Shakura & Sunyaev outer region characterizedby a Keplerian angular momentum profile. Since any global ac-cretion solution onto a black hole must cross the event horizonsupersonically in order to satisfy the fundamental boundary con-ditions imposed by general relativity, ADAF accretion disks arenecessarily transonic. Narayan et al. (1997) studied the behaviorof transonic ADAF disks subject to the Shakura–Sunyaev vis-cosity prescription. The associated asymptotic boundary condi-tions governing the dynamics close to the event horizon, and therelated physical restrictions on the viscosity prescription, wereexamined by Becker & Subramanian (2005) and Becker & Le(2003). Interest in the ADAF model remains strong, and it hasbeen the focus of many recent studies. For example, Narayan& McClintock (2008) have analyzed the properties of ADAFdisks close to the event horizon; Takahashi (2007) has exploredthe influence of causal viscosity prescriptions on the structureof ADAF disks; Ma et al. (2007) have probed the role of mag-netically induced torques in ADAF disks; and Yuan et al. (2008)have examined the properties of a simplified global model thatutilizes an approximate form for the radial momentum equation.

1.2. Critical Structure of Transonic Disks

Due to the supersonic nature of the inflow at the eventhorizon, any physically achievable flow configuration mustcontain at least one critical point, where the flow transitions fromsubsonic to supersonic. However, for a given set of incident flowparameters (e.g., specific energy, specific angular momentum,and viscosity), the flow may possess multiple critical points(Chakrabarti 1989a, 1989b, 1990, 1996; Chakrabarti & Das2004; Le & Becker 2005). This suggests the possible existence

649

650 DAS, BECKER, & LE Vol. 702

of global flow solutions with a standing shock wave locatedbetween two critical points. In their early study of viscous,transonic ADAF disks, Narayan et al. (1997) focused onglobal solutions that pass through a single critical point, andconsequently they did not obtain any shocked disk solutions.The problem of the structure of ADAF disks governed bythe Shakura–Sunyaev viscosity prescription was recently re-examined by Becker et al. (2008), who demonstrated thatisothermal shocks can exist in such disks even in the presenceof substantial viscosity.

In the present paper, we extend the approach taken by Beckeret al. (2008) by performing a complete analysis of the structureof viscous ADAF disks containing either single or multiplecritical points. Solutions with multiple critical points mayinclude a standing shock, if it is possible to satisfy the shockjump conditions at some radius r∗ between the outer criticalpoint (at radius r = rout

c ) and the inner critical point (at radiusr = r in

c ). Shocks occur when the accretion flow is impeded dueto the presence of a “centrifugal barrier” situated close to thehorizon. Although such barriers can exist even in viscous disks,they begin to dissipate as the level of viscosity increases becauseof the increasing efficiency of the outward diffusion of angularmomentum. Hence disks with large viscosity parameters are notexpected to possess shocks, and our results confirm this. In thepost-shock region, the flow accelerates toward the black hole,eventually passing through the inner critical point and crossingthe horizon supersonically.

1.3. Shock Waves in Viscous Disks

The question of the stability of viscous accretion flows con-taining standing shock waves is currently not resolved. How-ever, an important new development in the field was providedby Nagakura & Yamada (2008), who used both linear stabil-ity analysis and general relativistic hydrodynamical simulationsto investigate the stability properties of shock waves in invis-cid, advection-dominated accretion flows around Schwarzschildblack holes. When subjected to strong perturbations around theinitial condition, the authors find that the dynamical configu-ration is stable, in the sense that the shock continues to exist,although the shock radius may oscillate if the perturbationsare non-axisymmetric. Similar results were obtained by Okudaet al. (2007). Although these authors do not consider the effectof viscosity in their work, their conclusions nonetheless supportthe general idea that shocks may be able to exist in a stableconfiguration in relativistic disks. Consequently their findingsprovide a sound theoretical basis for the study of the structureof shocked, viscous disks developed in the present paper. Whena shocked solution is dynamically possible, we argue based onthe second law of thermodynamics that a shock will form sinceshocked-disk solutions possess higher entropy than smooth so-lutions (Becker & Kazanas 2001).

The study of shock waves in accretion flows around com-pact objects has been undertaken by a variety of authors us-ing either analytical methods (Fukue 1987; Chakrabarti 1989a,1989b, 1990, 1996; Abramowicz & Chakrabarti 1990; Nobuta& Hanawa 1994; Yang & Kafatos 1995; Yuan et al. 1996; Ca-ditz & Tsuruta 1998; Kovalenko & Lukin 1999; Das et al. 2001;Das & Chakrabarti 2004; Le & Becker 2004, 2005, 2007; Yuet al. 2006; Becker et al. 2008) or numerical simulation (Haw-ley et al. 1984a, 1984b; Chakrabarti & Molteni 1993; Molteni,Sponholz, & Chakrabarti 1996; Ryu et al. 1997; Chakrabartiet al. 2004). The physical configuration of the shock dependson the detailed microphysics of the dissipation mechanism op-

erating at the discontinuity. The three types of shocks mostfrequently studied are (1) Rankine–Hugoniot shocks, (2) isen-tropic shocks, and (3) isothermal shocks (Chakrabarti 1990;Abramowicz & Chakrabarti 1990). Rankine–Hugoniot shockspossess the largest post-shock temperatures because the energyflux is strictly conserved as the gas crosses the shock. Henceshocks of this kind are radiatively very inefficient. In an isen-tropic shock, the entropy generated via dissipation is imme-diately radiated away, and consequently such shocks displaysmaller temperature increases than the Rankine–Hugoniot vari-ety. In an isothermal shock, the temperature does not increase atall because all of the dissipated energy and entropy escape fromthe flow at the shock location. Hence isothermal shocks repre-sent the most efficient mechanism for removing energy from theflow.

Le & Becker (2004, 2005, 2007) demonstrated that theenergy removed from the flow at the isothermal shock locationcan be understood physically as a consequence of the first-order Fermi acceleration of relativistic particles in the disk,combined with the diffusion and escape of the acceleratedparticles through the upper and lower surfaces of the disk.This approach facilitates the development of a completelyself-consistent, coupled model that simultaneously describesthe dynamical structure of the disk/shock system and also theenergy spectrum of the relativistic particles escaping from thedisk to form the observed jet outflows.

The properties of isothermal shock waves in accretion diskshave been studied previously by Abramowicz & Chakrabarti(1990), Lu & Yuan (1998), Fukumura & Tsuruta (2004), andDas et al. (2003). However, the studies cited above all focused oninviscid flow models, which probably do not adequately describereal accretion disks. In this paper, we obtain for the first timedetailed dynamical solutions for the structure of ADAF diskscontaining isothermal shocks. The remainder of the paper isorganized as follows. In Section 2 we discuss the ADAF modelassumptions, the governing equations, and the critical conditionsfor the structure of ADAF disks. We discuss the isothermalshock jump conditions and examine global accretion solutionsin Section 3, and we explore the properties of the standing shockand classify the parameter space for shocked accretion solutionsin Section 4. Our dynamical formalism is used to model thedevelopment of the relativistic outflows in M87 and Sgr A∗ inSection 5, and concluding remarks are presented in Section 6.

2. GOVERNING EQUATIONS

We consider a thin, axisymmetric, viscous advection-dominated accretion disk structure. The disk is tenuous becauseit accretes matter at well below the Eddington rate, and thereforeenergy losses due to radiative processes are inefficient. Generalrelativistic effects are taken into account in an approximatemanner by utilizing the pseudo-Newtonian potential introducedby Paczynski & Wiita (1980). The self-gravity of the disk isneglected. Under these assumptions, there are two conservedquantities in viscous ADAF disks, namely the mass transportrate

M = 4πrHρ v, (1)

and the angular momentum transport rate

J = Mr2Ω − G, (2)

where r is the radial coordinate, H denotes the disk half-thickness, ρ is the mass density, v is the (positive) inflow

No. 1, 2009 DYNAMICAL STRUCTURE OF VISCOUS ACCRETION DISKS WITH SHOCKS 651

velocity, Ω is the angular velocity, and G denotes the torque.The energy transport rate is given by

E = −G Ω + M

(1

2v2 +

1

2v2

φ +P + U

ρ+ Φ

), (3)

where vφ = r Ω denotes the azimuthal velocity, P = (γ − 1) Uis the pressure, U is the internal energy density, and Φ representsthe pseudo-Newtonian potential, defined by (Paczynski & Wiita1980)

Φ(r) ≡ − GM

r − rS

, (4)

for a black hole with mass M and Schwarzschild radius rS =2GM/c2. This potential successfully reproduces many aspectsof the spacetime geometry around a nonrotating black hole. Weassume that the adiabatic index, γ , maintains a constant valuethroughout the flow, and the transport rates M , J , and E areall defined to be positive for inflow. Under our assumptions,M and J are conserved throughout the disk, and therefore theyrepresent the rates at which mass and angular momentum enterthe black hole, respectively. In general, the energy transport rateE is also conserved, except at the location of an isothermalshock if one is present in the flow.

The torque G is associated with the gradient of the angularvelocity Ω through the standard relation (e.g., Frank et al. 2002)

G(r) = −4πr3Hρ νdΩdr

, (5)

where ν is the kinematic viscosity. In the present work, we adoptthe Shakura & Sunyaev (1973) viscosity prescription,

ν(r) = αa2

ΩK, (6)

where α is a constant, ΩK denotes the Keplerian angular velocity,defined by

Ω2K(r) ≡ GM

r(r − rS )2= 1

r

dΦdr

, (7)

and a denotes the isothermal sound speed, given by

a2(r) = P

ρ. (8)

Combining Equations (1), (2), (5), and (6), we find that thegradient of the angular velocity can be written as

dΩdr

= −vΩK(� − �0)

αr2a2, (9)

where � ≡ r2Ω denotes the angular momentum per unit massfor the accreting gas at radius r, and �0 ≡ J /M is the angularmomentum transport rate per unit mass. Since the torque Gvanishes at the event horizon (Becker & Le 2003), it followsfrom Equation (2) that �0 is also equal to the specific angularmomentum of the material entering the black hole. The radialderivative of � is given by

d�

dr= 2 r Ω + r2 dΩ

dr, (10)

which can be combined with Equation (9) to obtain the differ-ential equation

d�

dr= 2 �

r− v�K(� − �0)

αa2r2, (11)

where the Keplerian specific angular momentum �K is definedby

�K(r) ≡ r2ΩK(r) = r3/2√

GM

r − rS

. (12)

The disk half-thickness H is estimated using the standardvertical hydrostatic relation

H (r) = ar2

�K. (13)

In a steady state, the comoving radial acceleration rate in theframe of the accreting gas can be written as

Dv

Dt≡ −v

dv

dr= 1

ρ

dP

dr+

�2K − �2

r3. (14)

By eliminating the torque G between Equations (2) and (3) andutilizing Equation (8), we find that the energy transport rate perunit mass is given by

ε ≡ E

M= 1

2v2 − 1

2

�2

r2+

�0�

r2+

γ

γ − 1a2 + Φ. (15)

Since we are working within the ADAF framework, ε willremain constant unless a shock is formed in the disk. In thisscenario, the variation of the internal energy density U isregulated by the adiabatic compression of the gas and the viscousdissipation in the disk. The comoving rate of change of U cantherefore be written in the frame of the gas as (e.g., Becker &Le 2003)

DU

Dt≡ −v

dU

dr= −γ

U

ρvdρ

dr+

ρν

r2

(d�

dr− 2�

r

)2

. (16)

We can obtain the dynamical differential equation govern-ing the spatial variation of the inflow velocity v by manipu-lating the conservation equations as follows. First we combineEquations (1), (13), and (8) to express the pressure as

P = Ma�K

4πr3v. (17)

Using this relation to substitute for the pressure in Equation (14)for the radial acceleration, we find that the derivative of the soundspeed a is given by

da

dr=

(a2

v2− 1

)v

a

dv

dr− a

�K

d�K

dr+

3 a

r+

�2 − �2K

ar3. (18)

Next we differentiate Equation (15) and combine the result withEquation (11) to obtain

dv

dr= �2 − �2

K

vr3− 2γ

γ − 1

a

v

da

dr− �K(� − �0)2

αa2r4. (19)

By using Equation (18) to substitute for da/dr in Equation (19),we obtain the differential dynamical equation (e.g., Narayanet al. 1997)(

v2

a2− 2γ

γ + 1

)1

v

dv

dr= 2γ

γ + 1

(3

r− 1

�K

d�K

dr

)

+

(γ − 1

γ + 1

)v�K(� − �0)2

αa4r4+

�2 − �2K

a2r3. (20)

In order to obtain the distributions of the flow variables v, a, and�, we must solve numerically the governing Equations (11), (15),and (20). This requires consideration of the critical structure ofthe flow as discussed below.

652 DAS, BECKER, & LE Vol. 702

2.1. Critical Conditions

A black hole will swallow matter either from its binarycompanion or from the winds of the surrounding stars. At theouter edge of the disk, the accreting matter has negligible radialvelocity, although it ultimately crosses the event horizon witha local velocity equal to the speed of light. Hence accretionflows around black holes must be transonic. The locations wherethe flow transitions from subsonic to supersonic are referredto as critical points. Depending on the parameters describingthe properties of the gas at a large distance from the blackhole, the flow may possess multiple critical points. When thisoccurs, it is interesting to investigate the possible existenceof isothermal standing shock waves located between two ofthe critical points. The location of the shock, if one exists, isdetermined by applying the shock jump conditions. We firststudy the critical structure of the transonic flow by rewritingEquation (20) in the equivalent form

dv

dr= N

D, (21)

where the numerator and denominator functions N and D aredefined by

N ≡ 2γ v

γ + 1

(3

2r+

1

r − rS

)

+

(γ − 1

γ + 1

)v2�K(� − �0)2

αa4r4+

v(�2 − �2

K

)a2r3

, (22)

and

D ≡ v2

a2− 2γ

γ + 1. (23)

In order to obtain well-behaved global solutions, the numer-ator and denominator functions must vanish at exactly the samelocation in the flow. This condition represents the mathematicaldefinition of a critical point. Setting N = D = 0, we obtain thecritical conditions

2γ vc

γ + 1

(3

2rc

+1

rc − rS

)+

(γ − 1

γ + 1

) [v2

c �K(�c − �0)2

αa4c r

4c

]

+vc

(�2

c − �2K

)a2

c r3c

= 0, (24)

andv2

c

a2c

− 2γ

γ + 1= 0, (25)

where rc, vc, ac, and �c denote the values of the radius, thevelocity, the sound speed, and the specific angular momentumat the critical point, respectively. Any accretion solution mustpass smoothly through the critical point, which is guaranteedif the critical conditions are satisfied. In this work, we assumeapproximate equipartition between the gas and magnetic pres-sures following Narayan et al. (1997), and accordingly we setγ = 1.5 in the subsequent analysis.

The topological nature of a critical point is determined by thevalue of the velocity gradient dv/dr at that point (e.g., Das 2007;Becker & Le 2003). In general, we obtain two possible valuesfor dv/dr , with one corresponding to accretion and the other toan outflow (wind). Any physically acceptable solution must passthrough a saddle type or X-type critical point which is obtained

when both of the derivatives are real and of opposite signs. Ifa shock forms, then the accretion flow must also pass throughanother saddle-type critical point in the post-shock region. Whenboth of the values of dv/dr are complex, the critical point isreferred to as O-type. This type of critical point is unphysicaland therefore no acceptable flow solution can pass through sucha point.

2.2. Inner Boundary Conditions

Solutions for the flow variables v(r), �(r), and a(r) areobtained by numerically integrating Equations (11) and (20),supplemented by the algebraic relation given by Equation (15).Becker & Le (2003) determined that in order to ensure thestability of the calculation, the integration must proceed inthe outward direction, starting from a point close to the eventhorizon. This requires the availability of asymptotic boundaryconditions that can be used to establish the starting values for thephysical variables. Based on the fundamental general relativisticprinciple that the torque must vanish at the horizon (Weinberg1972), Becker & Le (2003) derived the explicit asymptoticbehaviors of v(r), �(r), and the “entropy function,” K(r), formaterial subject to the Shakura–Sunyaev viscosity prescription,where

K(r) ≡ r v aγ +1γ−1

ΩK. (26)

The physical significance of K can be understood by combiningEquations (1), (13), (8), and (26) to show that

Kγ−1 ∝ P

ργ. (27)

This relation establishes that K remains constant in regions ofthe flow unaffected by dissipation, where P ∝ ργ .

If the gas is in local thermodynamic equilibrium, then onecan demonstrate that the value of K is related to the entropy perparticle S by (Reif 1965)

S = k ln K + c0, (28)

where c0 is a constant that depends only the composition ofthe gas, but is independent of its state. Becker & Le (2003)established that the asymptotic behaviors of K(r) and �(r) closeto the event horizon are given by

K(r) = K0

[1 − 16 α �2

0

5 c2r5S

( rS

2

)1/2(r − rS )5/2

], r → rS ,

(29)and

�(r) = �0 +2 α �0

c2rS

(2

rS

)1/2(

K20

2 r3S

) γ−1γ +1

(r − rS )γ +5

2γ +2 , r → rS ,

(30)where K0 represents the entropy value at the horizon. Notethat the asymptotic radial dependence of K(r) is very weak,reflecting the unimportance of viscous dissipation near thehorizon. Becker & Le (2003) also found that the asymptoticvariation of the inflow velocity is given by

v(r) = vff(r)

[1 +

2 ε0r2 − �2

0 − (γ + 1) f (r)

r2 v2ff(r) − (γ − 1) f (r)

]1/2

, r → rS ,

(31)

No. 1, 2009 DYNAMICAL STRUCTURE OF VISCOUS ACCRETION DISKS WITH SHOCKS 653

where ε0 denotes the value of the specific energy transport rateat the horizon, the free-fall velocity is given by

vff(r) =(

2 GM

r − rS

)1/2

, (32)

and the function f is defined by

f (r) ≡ 2 γ r2

γ 2 − 1

[K2

0

2 r3(r − rS )

] γ−1γ +1

. (33)

For given values of the parameters ε0, �0, and K0, Equations (30)and (31) can be used to compute starting values for theintegration of the variables �(r) and v(r) away from the eventhorizon.

3. GLOBAL SOLUTIONS

The structure of the global disk/shock model depends on theenergy transport rate at the horizon ε0, the angular momentumat the horizon �0, the entropy at the horizon K0, the viscosityparameter α, and the ratio of specific heats γ . In the calculationsperformed here, we set γ = 1.5, following Narayan et al. (1997).When a shock is present, we use the subscripts “−” and “+” todenote quantities measured just upstream and just downstreamfrom the shock, respectively. In a shocked disk, the gas passesthrough one critical point in the pre-shock region at r = rout

c ,and then through another in the post-shock region at r = r in

c .Conversely, if the flow is smooth (shock-free), then the gas onlypasses through one critical point, located at radius r = rc.

The process of determining the structure of a viscous ADAFdisk containing an isothermal shock begins with the selection ofinput values for ε0, �0, α, and γ . We must also select a provisionalvalue for the entropy parameter K0. Based on this information,we can utilize the asymptotic relations given by Equations (30)and (31) to compute values for �(r) and v(r), respectively,at the starting radius for the outward integration, which is2.001 GM/c2. Next we numerically integrate Equations (11),(15), and (20), beginning at the starting radius and workingoutward toward the inner critical point. The value of K0 is varieduntil a solution is obtained that satisfies the critical conditionsgiven by Equations (24) and (25), which ensure that the flowpasses smoothly through the inner critical point at radius r in

c .After the location of the inner critical point has been es-

tablished, the integration must be continued into the upstream,subsonic region. Due to the mathematical structure of the dy-namical Equation (21), it is not possible to integrate across thecritical point itself. We must therefore employ linear extrapo-lation, which requires the calculation of the derivative dv/drat the critical point using l’Hopital’s rule. After performing thelinear extrapolation, the integration is continued in the subsonicregion until we reach the shock radius, r∗, which is assigneda provisional value initially. Since no energy is lost from thedisk between the shock and the horizon, it follows that ε0 = ε+,where ε+ is the energy transport rate on the downstream sideof the shock. At the shock location, the velocity and the energytransport rate experience discontinuous transitions described bythe isothermal shock jump conditions, which can be written as(e.g., Chakrabarti 1989a)

v+

v−= M2

+ Δε ≡ ε+ − ε− = v2+ − v2

−2

, (34)

where M+ ≡ v+/a+ denotes the post-shock Mach number. Thisrelation facilitates the calculation of the pre-shock (incident)

Figure 1. (a) Inflow velocity v (solid lines) and isothermal sound speed amultiplied by

√2γ /(γ + 1) (dashed lines) plotted as functions of the radius r

in units of GM/c2 for a typical global accretion solution with γ = 1.5. Thethick and thin lines represent the shocked and smooth solutions, respectively.(b) Energy transport rate ε plotted as a function of radius r. Note the jump to thenegative post-shock value due to the release of energy through the upper andlower surfaces of the accretion disk at the isothermal shock location. See thetext for details.

energy transport rate, ε−. Note that if no shock is present, then εis continuous throughout the entire flow, and therefore Δε = 0and ε0 = ε−. By definition, the flow maintains a uniformtemperature across the isothermal shock, and consequentlya+ = a−. The shock radius r∗ is varied until we obtain a solutionthat passes smoothly through the outer critical point, locatedin the upstream region at radius rout

c . The solution process iscompleted when the critical conditions are satisfied at the outercritical point. For given input values of ε0, �0, α, and γ , in theend we obtain unique values for K0, r∗, r in

c , and routc , in addition

to the associated global dynamical solution for the disk/shockstructure.

A typical set of shocked and smooth accretion solutionsobtained for the same value of the incident energy transport rateε− is presented in Figure 1(a). In these calculations, the Shakura–Sunyaev viscosity parameter was specified by setting α = 0.1,and the specific heat ratio is given by γ = 1.5. By applyingthe multi-step iterative method described above, we obtain theparameters ε− = 0.001856, ε0 = ε+ = −0.003, �0 = 2.55, andK0 = 0.00689 for the shocked flow. In the shocked case, theincident flow first becomes supersonic upon passing through theouter critical point located at rout

c = 73.51. The gas subsequentlypasses through an isothermal shock at r∗ = 31.2, where thevelocity makes a discontinuous transition to become subsonicagain. Downstream from the shock, the flow accelerates towardsthe horizon, and regains its supersonic character after crossingthe inner critical point at r in

c = 6.798. The radius at theouter edge of the hot region, redge = 3555, is computed bysetting � = �K, so that the angular momentum matches thelocal Keplerian value. The radial profile of the specific angularmomentum � is plotted in Figure 2(b). Although the incidentenergy transport rate ε− is negative in our calculations, it shouldbe emphasized that the total energy inflow rate, including therest mass energy, is positive as required.

654 DAS, BECKER, & LE Vol. 702

Figure 2. Specific angular momentum � plotted as a function of the radius r for(a) α = 0.01, (b) α = 0.1, and (c) α = 0.2. Here we assume that γ = 1.5.The solid, dotted, and dashed lines correspond to the shocked, smooth, andKeplerian profiles, respectively. See the text for the complete sets of parametersused in each panel.

For the smooth-flow solution depicted in Figure 1(a), weused the parameters ε0 = ε+ = ε− = 0.001856, �0 = 2.368,and K0 = 0.008565. In this case, the value of �0 was selectedso that � = �K at the same outer edge radius, redge = 3555,obtained in the shocked disk (see Figure 2(b)). The smoothflow solution passes through a single critical point, located atrc = 8.097. It is interesting to compare the plots of the soundspeed obtained in the smooth and shocked cases. The loss ofenergy at the shock significantly reduces the sound speed (andtherefore the temperature) in the post-shock region relative tothe smooth solution. In Figure 1(b), we plot the variation of theenergy transport rate for the shock solution. The excess energyreleased through the upper and lower surfaces of the disk at theshock location may be sufficient to power the formation of arelativistic jet (Le & Becker 2005; Becker et al. 2008).

The viscosity parameter α plays a central role in determiningthe efficiency of the angular momentum transport in the disk. Weexplore the effect of varying this parameter in Figure 2, wherewe plot the results obtained for the radial profile of the specificangular momentum � in both shocked and smooth disks for threedifferent values of α. All of the solutions are sub-Keplerian(� < �K) for r < redge. In the shocked case, the increase inthe angular momentum in the post-shock region produces the“centrifugal barrier” required for shock formation, while no suchbarrier exists in the smooth case. In each of the panels, the valueof �0 was determined by satisfying the inner critical conditions,and the incident energy transport rate ε− has the same valuein the shocked and smooth scenarios. The parameter valuesadopted in panel (a) are α = 0.01, γ = 1.5, ε+ = −1.0 × 10−3,ε− = 1.9072 × 10−3, �0 = 3.0, K0 = 2.7509 × 10−3 in theshocked case, and ε+ = ε− = 1.9072 × 10−3, �0 = 2.845,K0 = 3.8588 × 10−3 in the smooth (shock-free) case. In panel(b), we adopt the same parameters used in Figure 1, namelyα = 0.1, γ = 1.5, ε+ = −3.0 × 10−3, ε− = 1.856 × 10−3,�0 = 2.55, K0 = 6.89 × 10−3 in the shocked case, andε+ = ε− = 1.856 × 10−3, �0 = 2.368, K0 = 8.565 × 10−3

in the smooth case. The values adopted in panel (c) areα = 0.2, γ = 1.5, ε+ = −1.0 × 10−3, ε− = 1.6483 × 10−3,�0 = 2.1, K0 = 1.1883 × 10−2 in the shocked case, andε+ = ε− = 1.6483×10−3, �0 = 1.966, K0 = 1.25×10−2 in thesmooth case. Note the increasing prominence of the centrifugalbarrier as α increases.

4. SHOCK PROPERTIES

Since this is the first work to extensively consider theconsequences of a standing shock for the structure of a viscousaccretion disk, it is interesting to examine how the shockproperties depend on the various flow parameters. We carry

Figure 3. Variation of (a) the isothermal shock radius r∗, (b) the compressionratio R = v−/v+, and (c) the energy jump Δε as functions of �0. Thesolid, dotted, dashed, dot-dashed, and short-long-dashed curves correspond toε0 = −0.006, −0.005, −0.004, −0.003, and −0.002, respectively. Here, we setα = 0.1 and γ = 1.5.

out the analysis for flows characterized by the fundamentalparameters (ε0, �0, α), where ε0 denotes the energy transportrate at the horizon, which is the same as the post-shock energytransport rate ε+. In each case, we keep one of the threeparameters (ε0, �0, α) fixed and vary the other two to studythe resulting shock properties.

In Figure 3, we present the results obtained for the shockradius, r∗, the compression ratio, R = v−/v+, and the energyjump, Δε = ε+ − ε−, as functions of the angular momentumtransport rate �0 and the energy transport rate at the horizon ε0.In these calculations, the viscosity and specific heat parametersare held constant, with α = 0.1 and γ = 1.5, respectively.Figure 3(a) clearly demonstrates that accretion flows with a verywide range of input parameters may possess isothermal standingshocks. Note that for a given post-shock energy parameter ε0, theshock location recedes away from the black hole with increasing�0 because of the enhancement of the centrifugal barrier. On theother hand, the shock location shifts inward as ε0 decreases for agiven value of �0, due to the increased ram pressure of the flow.For a given value of ε0, there is a restricted range of �0 valueswithin which a shock can exist. In general, we find that theshock radius r∗ always exceeds ∼10 GM/c2. In Figure 3(b), weplot the variation of the shock compression ratio R as a functionof the flow parameters ε0 and �0. We observe that for fixedε0, the compression ratio increases as the shock moves awayfrom the black hole. In Figure 3(c), we plot the correspondingvariation of the energy jump Δε. Note that Δε becomes morenegative as ε0 decreases for fixed �0, which reflects the fact thata larger amount of gravitational potential energy is released intothe outflow when the shock wave forms closer to the black hole.

Next we examine the dependence of the shock properties onthe viscosity parameter α. In Figure 4(a), the isothermal shocklocation r∗ is plotted as a function of α for various values of the

No. 1, 2009 DYNAMICAL STRUCTURE OF VISCOUS ACCRETION DISKS WITH SHOCKS 655

Figure 4. Variation of (a) the isothermal shock radius r∗, (b) the compressionratio R = v−/v+, and (c) the energy jump Δε as functions of α. The solid, dotted,and dashed lines denote ε0 = −0.006, −0.005, and −0.004, respectively. Here,we set �0 = 2.62 and γ = 1.5.

post-shock (horizon) energy transport rate ε0 for the case withγ = 1.5. Here, we fix the angular momentum transport rate bysetting �0 = 2.62. In all cases, we observe that shocks can formover a wide range of flow parameters. For a given α, the shocktends to form farther away from the black hole as ε0 is increased,which is consistent with the behavior noted in Figure 3(a). Wealso observe that for fixed ε0, the shock radius r∗ increases withα. This is because higher values of α tend to reduce the angularmomentum, which forces the centrifugal barrier to operate at alarger radius in order to be effective. Note that for a given setof flow parameters (ε0, �0), the viscosity parameter α has bothupper and lower limits for shock formation. The lower limitreflects the fact that fully inviscid flows cannot form shocks forarbitrary combinations of ε0 and �0 (e.g., Le & Becker 2005).Conversely, the upper limit on α is due to the efficient dissipationof angular momentum that occurs as α increases. This effectprevents the development of the centrifugal barrier required forshock formation if α is too large.

The dependence of the shock properties on the post-shockenergy ε0 is illustrated in Figure 5 for three values of theangular momentum transport rate �0. The viscosity and specificheat parameters are held constant with the values α = 0.1 andγ = 1.5, respectively. In Figure 5(a) we plot the shock locationr∗ as a function of ε0. Since the standing shock is centrifugallysupported, the shock radius r∗ increases when �0 is increased fora fixed value of ε0. On the other hand, if we hold r∗ constant, thenwe observe an anticorrelation between �0 and ε0. In Figures 5(b)and (c), respectively, we depict the associated variations of thecompression ratio R and the energy jump Δε as functions of ε0for the same set of flow parameters used in Figure 5(a). Thecompression ratio increases when either �0 or ε0 is increased.We note that Δε increases almost linearly with increasing ε0 forfixed �0. The results indicate that flows with larger values of �0tend to discharge less energy at the shock location.

Figure 5. Variation of (a) the isothermal shock radius r∗, (b) the compressionratio R = v−/v+, and (c) the energy jump Δε as functions of ε0. The solid,dotted, and dashed curves represent �0 = 2.62, 2.67, and 2.72, respectively.Here, we set α = 0.1 and γ = 1.5.

The hot, sub-Keplerian ADAF region must connect with acool Keplerian outer disk at some large radius. We estimate theouter edge radius of the ADAF zone, redge, by setting the angularmomentum of the accretion flow equal to the local Keplerianvalue. In Figure 6, we plot the variation of the outer edge radiusas a function of �0 and ε0 for the case with α = 0.1 andγ = 1.5. We observe that for a fixed value of ε0, the valueof redge rapidly increases with increasing �0, indicating that thecool Keplerian disk recedes from the horizon. The outer edgeradius also increases with increasing ε0 when �0 is held fixed.

The shocked, viscous, advection-dominated disks discussedhere represent a new type of astrophysical phenomenon, andtherefore it is important to investigate the range of flow param-eters within which shocks can form. In Figure 7, we analyzethis question for the case with γ = 1.5 by varying the angularmomentum transport rate �0 and the upstream (incident) energytransport rate ε− while holding the viscosity parameter α fixed atfive different values. The solid curve depicts the region withinwhich shocked solutions are possible for fully inviscid flow(α = 0). This result agrees very well with Figure 2 from Le &Becker (2005). We note that the allowed region for the shockedsolution shifts toward lower angular momentum as α increases.This shift occurs because the specific angular momentum of theaccreting material at the horizon, �0, decreases in response tothe enhanced viscosity associated with the increase in α. Theshrinkage of the region for the shocked solution observed asα increases suggests that shocked disk solutions are impossi-ble above a critical value of α, denoted by αcrit. For the modelconsidered here, with γ = 1.5, we find that αcrit ≈ 0.27.

5. ASTROPHYSICAL APPLICATIONS

It is interesting to ask whether the self-consistent disk/shock model we have described can account for the energetics

656 DAS, BECKER, & LE Vol. 702

Figure 6. Plot of the outer edge radius redge as a function of �0. This isthe radius where the angular momentum � equals the Keplerian value �K(Equation (12)). The solid, dotted, and dashed lines correspond to ε0 = 0.0,−0.004, and −0.008, respectively. Here, we set α = 0.1 and γ = 1.5. See thetext for details.

Figure 7. Representation of the parameter space spanned by the specific angularmomentum at the horizon �0 and the incident energy transport rate ε− whenγ = 1.5. The wedge-shaped regions denote the areas of the parameter spacewithin which shocks can form for the indicated values of α. See the discussionin the text.

of specific jet sources, such as M87 and Sgr A∗. Whenobservational values of the black hole mass M, the accretionrate M , and the jet kinetic luminosity Ljet are available for aparticular source, we can utilize this information to constrainthe model parameters. The process begins by selecting valuesfor the fundamental free parameters ε0, α, and γ . After selectingprovisional values for �0 and K0, we utilize the asymptoticrelations given by Equations (30) and (31) to compute initial

values for the functions �(r) and v(r) at the starting radius2.001 GM/c2. We then numerically integrate Equations (11),(15), and (20) in the outward direction, and vary the value ofK0 until the critical conditions given by Equations (24) and (25)are satisfied at the inner critical point.

The energy transport rate ε drops from its upstream value ε−to the downstream value ε+ = ε0 at the shock location due tothe escape of energy from the disk into the outflow (jet). Wecan therefore determine the shock radius r∗ by requiring thatthe value of Δε computed using Equation (34) agrees with thatcalculated based on the observed jet kinetic luminosity, Ljet,using the relation

Ljet = −MΔε, (35)

where the negative sign appears because Δε < 0. Oncethe shock radius is established, the integration is continued onthe upstream side of the shock toward the outer critical point.If the flow does not pass smoothly through the outer criticalpoint, then the value of �0 is modified and the procedure isrepeated starting from the horizon. The computation of the self-consistent disk/shock dynamical model is complete when thecritical conditions are satisfied at the outer critical point. Forgiven input values of ε0, α, and γ , in the end we obtain thecomplete global dynamical solution along with unique valuesfor �0, K0, r∗, r in

c , and routc . We provide two specific examples

below.

5.1. M87

The disk/shock/outflow model we have developed was previ-ously utilized by Becker et al. (2008) to analyze the structure ofthe accretion flow in M87. The results obtained for that sourceare presented in more detail here. Our application to M87 isbased on the observational values M = 1.34 × 10−1 M yr−1

(Reynolds et al. 1996), Ljet = 5.5 × 1043 erg s−1 (Reynoldset al. 1996; Bicknell & Begelman 1996; Owen et al. 2000), andM = 3.0 × 109 M (e.g., Ford et al. 1994). We set the vis-cosity parameter α = 0.1 in order to demonstrate that shockscan exist in ADAF disks even in the presence of substantialviscosity, although our model can accommodate any value forα. Based on the assumption of approximate energy equiparti-tion between the magnetic field and the gas internal energy, weadopt the value γ = 1.5 for the adiabatic index (Narayan et al.1997). We utilize natural gravitational units in our numericalexamples, with GM = c = 1 and rS = 2, except as noted.The remaining parameters for the shocked-disk model can becomputed based on the observations of M87, from which weobtain ε− = 0.001516, ε+ = ε0 = −0.005746, �0 = 2.6257,K0 = 0.00608, r∗ = 26.329, r in

c = 6.462, and routc = 96.798.

For the pre-shock and post-shock velocities and Mach num-bers we obtain v− = 0.138, v+ = 0.068, M− = 1.427, andM+ = 0.701, respectively, and for the disk half-thickness at theshock location we obtain H∗ = 12.10.

The results obtained for the inflow speed v and the isothermalsound speed a are plotted in Figure 8(a), and the associatedsolution for the specific angular momentum � is depicted alongwith the Keplerian profile �K (Equation (12)) in Figure 8(b).Results are presented for both shocked and smooth (shock-free)disks. In the shocked disk, the source is located at the shockradius, r∗ = 26.329, and we observe that the values of � and �Kmerge at redge = 4658, which represents the outer edge of theADAF region. The smooth solution is computed by integratingthe conservation equations using the same value for the incidentenergy transport rate ε− as adopted in the shocked-disk model,

No. 1, 2009 DYNAMICAL STRUCTURE OF VISCOUS ACCRETION DISKS WITH SHOCKS 657

Figure 8. Dynamical profiles for M87. (a) Inflow velocity v (solid lines) andisothermal sound speed a multiplied by

√2γ /(γ + 1) (dashed lines) plotted as

functions of the radius r in units of GM/c2. The thick and thin lines denote theshocked and smooth solutions, respectively. (b) Specific angular momentum �

for the shocked (thick line) and smooth (thin line) solutions plotted as functionsof r in units of GM/c2 along with the Keplerian angular momentum �K (dot-dashed line).

but with no shock included. The value of �0 is then varied untilwe obtain � = �K at the same outer radius, redge = 4658, as inthe shocked disk. The resulting parameter values for the smoothsolution are ε− = ε+ = 0.001516, �0 = 2.3988, K0 = 0.0084,and rc = 7.572. In order to facilitate comparison with theshocked model, in the smooth case the source is assumed tobe at the same radius r = 26.329.

Our solutions for the specific angular momentum � are sig-nificantly sub-Keplerian, in agreement with the results obtainedby Narayan et al. (1997), who utilized the same set of ADAFconservation equations we employ. The sub-Keplerian nature ofthe flow stems from the relatively high value for the viscosityparameter adopted here, α = 0.1. Note that the sound speed (andhence the temperature) is significantly lower in the shocked caseas compared with the smooth disk due to the release of energyinto the outflow at the shock location.

It is important to examine the energy balance in the shockedmodel to ensure that the rate of energy loss at the shock locationis equal to the observational value for the kinetic luminosityin M87. We can compute the theoretical value for the jetluminosity, Ljet, by substituting our values for ε+ and ε− intoEquation (35). The result obtained is Ljet = 5.5 × 1043 erg s−1,which agrees with the observed jet luminosity for M87. Thisconfirms that our viscous shocked disk model is able to accountfor the energetics of the M87 outflow.

5.2. Sgr A∗

For Sgr A∗, we adopt the values M = 2.6×106 M (Schodelet al. 2002) and M = 8.96 × 10−7 M yr−1 (Yuan et al.2002; Quataert 2003), and we set γ = 1.5 and α = 0.1.The kinetic luminosity of the jet is estimated using Ljet =5 × 1038 erg s−1 (Falcke & Biermann 1999), although thisvalue is rather uncertain (e.g., Yuan 2000; Yuan et al. 2002).The results for the remaining model parameters implied bythe observations are ε− = 0.00134884, ε+ = ε0 = −0.0085,�0 = 2.6728, K0 = 0.005448, r∗ = 19.917, r in

c = 6.380, androutc = 112.384. For the pre-shock and post-shock velocities

Figure 9. Same as Figure 8, except model results are for Sgr A∗.

and Mach numbers we obtain v− = 0.159, v+ = 0.0748,M− = 1.4578, and M+ = 0.6860, respectively, and for thedisk half-thickness at the shock location we obtain H∗ = 8.72.

In Figure 9(a), we plot the results obtained for the inflowspeed v and the isothermal sound speed a in both the shockedand smooth disk models for Sgr A∗. The profiles of � and �K areplotted in Figure 9(b). In the shocked case, the source is locatedat r∗ = 19.917, and we find that � = �K at radius redge = 5432,which is the outer edge of the ADAF region. The incident energytransport rate ε− = 0.00134884 is the same for both the smoothand shocked models, and the value of �0 in the smooth case isdetermined by requiring that � merges with the Keplerian profileat the same radius redge = 5432 as in the shocked case. Based onthis approach, the parameters obtained for the smooth model areε− = ε+ = 0.00134884, �0 = 2.416, K0 = 0.008293, and rc =7.511. We assume for consistency that the source is located at thesame radius r = 19.917 in both the smooth and shocked cases.

The results for the specific angular momentum � are sig-nificantly sub-Keplerian, as expected based on the relativelyhigh value utilized for the viscosity parameter, α = 0.1 (cf.Figure 8(b)). The sound speed is lower when a shock is present,due to the escape of energy at the shock location, as noted previ-ously in Figure 8(a). The result for Ljet obtained by substitutingour values for ε+ and ε− into Equation (35) is equal to the ob-served jet luminosity, 5 × 1038 erg s−1, and this establishes thatour shocked disk model successfully accounts for the observedenergetics of the accretion flow and the outflow in Sgr A∗.

6. CONCLUSION

In this paper we have presented the first systematic studyof the structure of stationary, advection-dominated, shockedviscous accretion disks around black holes. Our primary focushere has been on the global properties of accretion flowscontaining isothermal shock waves, formed as a result ofthe presence of a centrifugal barrier located near the eventhorizon. Our model is based on the same set of standard ADAFconservation equations considered by Narayan et al. (1997). Theconservation equations are supplemented by the inner boundaryconditions developed by Becker & Le (2003) which allow thecalculation of starting values for the physical variables close tothe horizon. Once the starting values are established, a stableoutward integration method is employed to obtain the radial

658 DAS, BECKER, & LE Vol. 702

profiles of the various flow quantities such as velocity, soundspeed, and angular velocity. The fundamental model parametersare the energy inflow rate at the horizon, ε0, the specific angularmomentum of the material at the horizon, �0, the entropyparameter at the horizon, K0, the Shakura–Sunyaev viscosityparameter, α, and the ratio of specific heats, γ .

For given observational values of the black hole mass M,the accretion rate M , and the jet kinetic luminosity Ljet, weare free to select values for α, γ , and ε0. Following Narayanet al. (1997), we have set γ = 1.5 in the calculations presentedhere. The remaining parameters �0 and K0, and the shock radiusr∗, are subsequently determined by applying the asymptoticinner boundary conditions given by Equations (30) and (31),the critical conditions given by Equations (24) and (25), andthe shock jump conditions given by Equations (34) and (35).The dynamical structure of the disk/shock system is obtainedby numerically integrating Equations (11), (15), and (20) in theoutward direction beginning at a starting radius located close tothe event horizon.

The formation of a shock requires that the flow possessesat least two critical points. The existence of multiple criticalpoints in both adiabatic and polytropic accretion flows hasbeen reported by many authors (e.g., Abramowicz & Zurek1981; Chakrabarti 1989a, 1989b, 1990, 1996). Flows withmultiple critical points are confined to a restricted region ofthe parameter space, as indicated in Figure 7. Accretion flowswithin this region can form isothermal shocks. In their earlierstudy, Narayan et al. (1997) focused solely on solutions that passonly through the inner critical point, and consequently they didnot obtain shocked disk solutions. Flows with discontinuitiesmust first pass through an outer critical point, and subsequentlythrough a shock and an inner critical point in order to representphysically acceptable accretion solutions. That is the scenariowe have investigated here.

The possible presence of shock waves in accretion disks andtheir observational implications has been examined by a varietyof previous authors. Abramowicz & Chakrabarti (1990) pre-sented one of the earliest studies of the properties of standingshock waves in accretion disks. Subsequently, the particular im-portance of isothermal shock waves was realized by Le & Becker(2004, 2005, 2007), who pointed out that efficient particle accel-eration in the vicinity of a standing isothermal shock can helpto explain the formation of the relativistic outflows of mattercommonly observed around radio-loud, underfed black holes.However, all of these previous studies of isothermal shocks inaccretion disks neglected the important role of viscosity in de-termining the disk structure. Very recently, Becker et al. (2008)demonstrated that first-order Fermi acceleration inside a vis-cous, shocked ADAF can accelerate particles more efficientlythan in a smooth disk. Hence the presence of a standing shockwave can create a potentially favorable environment for the pro-duction of relativistic outflows. In this paper, we have presenteda detailed study of the dynamics of viscous, shocked ADAFdisks that are governed by the standard description of the an-gular momentum transport based on the Shakura–Sunyaev pre-scription for the kinematic viscosity. In particular, we confirmthat the energetics of the viscous disk/shock model can explainthe observational properties of both the accretion flow and therelativistic outflows in M87 and Sgr A∗.

In Figure 7 we explored the parameter space within whichADAF disks with isothermal shocks can exist as a functionof the specific angular momentum at the horizon �0 and theincident energy transport rate ε− for various (constant) values of

the viscosity parameter α. The shock region shifts toward lowervalues of �0 as α increases due to the enhanced outward diffusionof angular momentum. We find that for the canonical valueγ = 1.5, shocked disk solutions can exist provided α � 0.27.This establishes for the first time that highly viscous ADAFdisks can contain standing shock waves. The stability of such ashock is an open question, but Nagakura & Yamada (2008) andOkuda et al. (2007) recently found that shocks in inviscid disksare persistent, although the shock radius may oscillate. Theseresults at least suggest the possibility that shocks in viscous disksmay be stable as well, although this needs to be investigated infuture work. Since shocked solutions are expected to possesshigher entropy than smooth solutions, we argue based on thesecond law of thermodynamics that when shock solutions arepermitted dynamically, they should be expected to form (Becker& Kazanas 2001).

Although Gu & Lu (2001, 2004) obtained solutions for ADAFdisks containing Rankine–Hugoniot shocks, the results devel-oped here represent the first dynamical solutions for ADAF diskswith a significant level of viscosity (i.e., α = 0.1) containingisothermal shocks. The Rankine–Hugoniot shocks studied byGu & Lu (2001, 2004) have conserved energy transport rates,and therefore they cannot directly produce relativistic outflows(jets). Conversely, the isothermal shocks we study here have adiscontinuous energy transport rate and consequently they areable to power the observed outflows. In our analysis of the diskstructure, we have assumed that the accretion rate M remainsconstant throughout the disk, and therefore we have neglectedthe effect of the outflow mass loss on the dynamics of the disk,despite the fact that an outflow emanates from the shock loca-tion in our model. This assumption is probably reasonable sinceestimates show that the rate of mass loss from the shock into thewind is typically ∼10−3 M if the jet is hadronic (Le & Becker2005); this ratio is further reduced by the factor me/mp if thejet is leptonic. However, there are two other potential forms ofmass loss from the disk that must also be considered. The firstis the possibility of hydrodynamically driven outflows of thebackground thermal gas, and the second is the possibility ofthe escape of a significant number of relativistic particles fromthe inner and/or outer regions of the disk, rather than from theshock itself. We consider each of these issues below.

Narayan et al. (1997) and Blandford & Begelman (1999) havenoted that ADAF disks typically possess a positive Bernoulliparameter,

B(r) ≡ v2

2+

�2

2 r2+

γ a2

γ − 1− GM

r − rS

, (36)

suggesting that sufficient energy is available to power hydro-dynamical outflows of the thermal gas. It is interesting tore-examine this issue in the context of the shocked, viscousaccretion disks considered here. In Figures 10(a) and (b),respectively, we plot the Bernoulli parameter as a function ofradius based on the smooth and shocked models for M87 andSgr A∗ developed in Section 5. Both the smooth and shockedsolutions display a slightly positive Bernoulli parameter in theinner and outer regions, suggesting that the gas is unbound there.When a shock is present, the gas is bound in the post-shock re-gion, but it eventually becomes unbound again before crossingthe horizon.

The Bernoulli parameter is related to the energy transport rateE and the viscous torque G via (see Equation (3))

E = −G Ω + MB. (37)

No. 1, 2009 DYNAMICAL STRUCTURE OF VISCOUS ACCRETION DISKS WITH SHOCKS 659

Figure 10. Bernoulli parameter (Equation (36)) plotted as a function of theradius r in units of GM/c2 for (a) M87 and (b) Sgr A∗. The thick and thin linesdenote the shocked and smooth solutions, respectively.

Since E is conserved in the inner region, it follows that the posi-tive bump in the Bernoulli parameter between the shock and thehorizon reflects an increase in the torque-driven outward energytransport rate, G Ω. This effect is less pronounced in the shockedsolution because the escaping particles carry away a significantfraction of the binding energy. The positivity of the Bernoulliparameter in the inner and outer regions suggests the possibil-ity of hydrodynamical outflows emanating from the non-shockregions of the flow. However, any such outflows would be quiteweak, with terminal Lorentz factors Γ = B + 1 ∼ 1.01, whereasthe terminal Lorentz factor for the outflow from the shock isΓ ∼ 6 (Becker et al. 2008; P. A. Becker et al. 2009, in prepara-tion). More fundamentally, it must be emphasized that despitethe positivity of the Bernoulli parameter, there is no hydrody-namical mechanism in the model considered here capable ofgenerating an outflow, since the ADAF disks we focus on arein vertical hydrostatic equilibrium, and therefore the net verti-cal force vanishes (cf. Equation (13)). Hence some additionalmechanism, outside the standard ADAF framework, must beoperative in order to channel the energy into the outflows. Inour view, the outflows are not hydrodynamical in nature, butare instead powered by the first-order Fermi acceleration ofrelativistic particles in the vicinity of the discontinuous shock.

In addition to the possibility of hydrodynamical mass loss,some mass may also be lost from the disk in the form of rela-tivistic particles that escape from the inner and outer regions ofthe disk, rather than from the shock itself. The escape of theseadditional relativistic particles could be driven by the distributedfirst-order Fermi acceleration occurring throughout the disk asa consequence of the overall compression of the flow. The sig-nificance of this effect can only be understood by performinga detailed study of the relativistic particle transport occurringthroughout the disk, which is beyond the scope of this paper.We have developed such a model and will report the results in aseparate paper. Preliminary calculations suggest that the spatialdistribution of the escaping particles is strongly peaked at theshock location, with about half of the particles escaping from theshock itself, and the remainder escaping from the surroundingregion. Hence, we conclude that the loss of mass from the diskis strongly concentrated in the vicinity of the shock, and that itis dominated by the escape of relativistic particles, rather thanby hydrodynamical outflows of the background (thermal) gas.

The authors thank the anonymous referee for providingseveral useful comments that significantly strengthened thepaper. S.D. acknowledges support via a postdoctoral fellowshipfrom the Korea Astronomy and Space Science Institute (KASI).

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